Convex Chance-Constrained Stochastic Control under Uncertain Specifications with Application to Learning-Based Hybrid Powertrain Control

Teruki Kato; Ryotaro Shima; Kenji Kashima

eess.SYJan 26, 2026

Convex Chance-Constrained Stochastic Control under Uncertain Specifications with Application to Learning-Based Hybrid Powertrain Control

Authors: Teruki Kato, Ryotaro Shima, Kenji Kashima

Organizations: Toyota Central R&D Labs., Inc., Nagakute, Aichi, Japan · Graduate School of Informatics, Kyoto University, Kyoto, Japan

Abstract

This paper proposes a strictly convex chance-constrained stochastic control framework that accounts for uncertainty in control specifications such as reference trajectories and operational constraints. Control inputs and risk allocation are jointly optimized under general, possibly non-Gaussian, uncertainties through sufficient deterministic conditions for probabilistic constraint satisfaction. A decreasing risk regularizer ensures existence and full use of the allowable risk. Under the stated regularity assumptions, the joint solution is unique and locally Lipschitz-continuous, supporting stable numerical optimization. The formulation is extended to learning-based exactly linearizable (EL) models through monotone state and input transformations. A hybrid powertrain example compares deterministic control, equal risk allocation, and optimized risk allocation with and without regularization in terms of charge sustaining, constraint violations, closed-loop cost, and solver time.

Figures & tables

Method	Specification uncertainty	Chance constraints	Non-Gaussian	Strict convexity	Nonlinear dynamics
[ 7 ]	$\times$	✓	✓	$\times$	$\times$
[ 8 ]	$\times$	✓	$\times$	$\times$	✓
[ 9 ]	$\times$	✓	$\times$	$\times$	$\times$
[ 3 ]	$\times$	$\times$	✓	✓	$\times$
[ 12 , 13 , 14 ]	✓	✓	$\times$	$\times$	✓
[ 15 ]	✓	$\times$	✓	✓	$\times$

TABLE I: Comparison with existing studies

Item	Value
Sampling time	$0.1$ s
Prediction horizon length $N$	$10$
Number of simulation steps	$447$ ( $44.7$ s)
Initial state $[\tau_{0}^{\mathrm{eng},\mathrm{ds}},V_{0},Q_{0}]$	$[50\ \mathrm{Nm},V_{0}^{\mathrm{req}},30\ \mathrm{Ah}]$
Speed-tracking weight $w^{\mathrm{req}}$	$10$
Engine-command weight $w^{\mathrm{eng}}$	$1.0\times 10^{-3}$

TABLE II: Simulation settings

Fig. 3: Horizon-summed risk allocations and empirical violation probabilities.

Control method	Solver time [ms] Mean $\pm$ SD	Maximum time [ms]	Closed-loop cost (total)	$Q_{T}-Q_{0}$ [Ah]
Deterministic	$15.407\pm 0.829$	$18.088$	$28.6566$	$-1.284$
Equal risk	$16.062\pm 1.095$	$19.372$	$122.4649$	$+1.822$
Optimized risk (regularized)	$21.999\pm 2.945$	$29.541$	$31.6265$	$+0.739$
Optimized risk (unregularized)	$29.542\pm 7.313$	$109.605$	$35.6778$	$+0.432$

TABLE III: Solver time and closed-loop performance

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Chance-Constrained Optimization Markov Decision Processes Stochastic Optimal Control Constrained RL

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In this paper, we consider stochastic optimal control problems with infinite-horizon joint chance constraints. By means of an appropriate state augmentation, we reformulate the original problem as a constrained Markov decision process, in which both the cost and the constraint function exhibit an additive structure. We then prove that this formulation enjoys strong duality, thereby enabling us to reformulate the problem as an equivalent unconstrained one in the Lagrange dual framework. We propose a dual-ascent algorithm to solve the resulting problem and show that it converges to a deterministic Markov policy defined over the augmented state space that is both optimal and feasible. To accommodate continuous state-input spaces, we propose a dedicated learning algorithm to approximate the value function in an offline training setting, thereby significantly reducing the computational complexity of the online control phase. We then test our approach on a numerical example and demonstrate its effectiveness compared to online predictive control methods in terms of performance and computational complexity.

Abstract

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